C. Solve what is asked: 1. Insert four arithmetic means between -1 and 14. 2. Insert five arithmetic means between 14 and 86. 3. Insert three arithmetic means between -16 and 4.
Answer (1)
Answer:To insert arithmetic means between two numbers, we use the Arithmetic Sequence formula:a_n = a_1 + (n - 1)dWhere: = first term = last term = total number of terms (including means and endpoints) = common difference---1. Insert four arithmetic means between -1 and 14That means a total of 6 terms:-1,\ \_\_,\ \_\_,\ \_\_,\ \_\_,\ 14So:a_1 = -1,\quad a_6 = 14,\quad n = 6Use formula:a_6 = a_1 + (6 - 1)d \\14 = -1 + 5d \\15 = 5d \Rightarrow d = 3Now find the means:2nd term: 3rd term: 4th term: 5th term: ✅ Answer: 2, 5, 8, 11---2. Insert five arithmetic means between 14 and 86We need 7 terms:14,\ \_\_,\ \_\_,\ \_\_,\ \_\_,\ \_\_,\ 86Given:a_1 = 14,\quad a_7 = 86,\quad n = 7a_7 = a_1 + 6d \Rightarrow 86 = 14 + 6d \\72 = 6d \Rightarrow d = 12Now compute the means:2nd: 3rd: 4th: 5th: 6th: ✅ Answer: 26, 38, 50, 62, 74---3. Insert three arithmetic means between -16 and 4Total of 5 terms:-16,\ \_\_,\ \_\_,\ \_\_,\ 4a_1 = -16,\quad a_5 = 4,\quad n = 54 = -16 + 4d \Rightarrow 20 = 4d \Rightarrow d = 5Now find the means:2nd: 3rd: 4th: ✅ Answer: -11, -6, -1
